We must express Newton's laws of motion in terms of four- vectors so
that they are frame invariant and consistent with Special Relativity.
The four- velocity
of a particle is the tangent to its worldline of length c [ see
Figure 2.1 ]:

Figure 2.1: The worldline of a particle with four- velocity .

This is the most natural analogue of the three- velocity. It is clearly
a four- vector since both and are invariant.

In the particle's own rest frame , the four- velocity is

It follows that in a general frame :

or

where is the particle's three- velocity.

For low velocities , and the spatial part is
nearly the same as .
If an observers own four- velocity is written as
in we have